26 - Equilibrium fluctuations and directed polymers

Summary

In Chapter 9 we discussed the dynamics of a driven interface in a porous medium. However, there are a number of problems in physics in which one is interested in the properties of an equilibrium interface, when there is no driving force pushing the interface in a selected direction. A closely related problem is the equilibrium fluctuations of an elastic line in a porous medium, which is a problem of interest in many branches of physics ranging from flux lines in a disordered superconductor (see Fig. 1.4) or the motion of stretched polymer in a gel. In this chapter we discuss the properties of a directed polymer (DP) in a two-dimensional random medium, and the relation between the directed polymer problem and the interface problem.

Discrete model

Consider a discrete lattice whose horizontal axis is x, and vertical axis is h (see Fig. 26.1). The polymer starts at x = 0 and h = 0, and moves along the x direction in discrete steps. It can go directly along x, or it can move via transverse jumps, such that |h(x + 1) – h(x)| = 0, 1. There is an energy cost (‘penalty’), for every transverse jump. This simulates a line tension, discouraging motion along the h axis. On every bond parallel to x, a random energy ε(x, h) is assigned.

The total energy of the DP is the sum over the energies along the polymer length, which includes the sum over the random energies arising from motion along the x direction, and the energy penalties for motion along the h axis.